Definitions are the core of mathematical precision; they come to answer "what is X" in mathematics. Into this category fit questions regarding equivalence of definitions; clarifications regarding complicated definitions; as well questions with purposed definitions for mathematical notions with ...

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Are equidimensional ideals and unmixed ideals the same?

Zariski-Samuel define an unmixed/equidimensional ideal to be one whose associated primes have the same dimension. At other places I have seen definitions saying unmixed=all associated primes have same ...
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Definition of stereoprojection and Möbius maps

@WillieWong has kindly pointed out that there are 2 definitions of stereographic projection. One with the unit sphere placed on top of the plane, the other where the plane is at the equator of the ...
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Is the extra condition in this definition superfluous?

I am learning Differential Geometry and someone told me that the second condition of a definition provided in books follows from the first and is hence superfluous. I cannot dispute it, so convince me ...
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Definition of quotient category

Is there any reason why only gluing of morphisms sharing domain and codomain is usually allowed in the definition of quotient category?
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Fountain code and online code

Are fountain code and online code the same? It seems to me they have the same property, which is used in lossy channel and generate unlimited encoded block. If they are the same, then what encoding ...
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60 views

Confused by definition of an open set in “All the Mathematics You Missed”

On page 66 of Thomas Garrity's "All the Mathematics You Missed", Garrity gives the following definition of an open set in $\mathbb{R}^n$: A set $U$ in $\mathbb{R}^n$ will be open if given any $a ...
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Conjugate actions

What is the definition for two group actions to be conjugate? For example a smooth action of a finite group on a manifold is locally conjugate to an orthogonal action.
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Definition of Non-characteristic Manifold

What is the definition of non-characteristic manifold in the following context. "Since $ f (x, y)=0 $ is assumed to be a non-characteristic singularity manifold, we have $ f_{x}\neq 0 $." Thanks, ...
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Vanishes of second order

I encountered the following statement in an article: Let $f(X) = f(X_1, ..., X_N) = \sum_{|\alpha| \geq 2} {a_{\alpha}X^{\alpha}}$ be a convergent power series which vanishes of second order at ...
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How do you call a scale that starts at $∞$, has $1/n$ divisions and tends to $0$?

A linear scale $2n$ divisions: 0 2 4 6 8 Logarithmic scales $10^n$ divisions: 1 10 100 1000 10000 ...
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30 views

Notation problem with a set of tuples and a metric

The first question: Assume we have tuples $T_i = (x_i, \vec{c}_i)$ ($x_i$ is the name of the object which is characterized by $\vec{c}_i$ in a d-dimensional space) and define a set of them $TS = ...
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Name for “relative difference”

If we want to express that two numbers $x, y$ are not so far away from each other absolutely we use the absolute value function $|\cdot|$ with $0 < \epsilon \ll 1$: $$|x-y| < \epsilon$$ ...
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Category of a PDE and its properties

Now I am working on numerical method for a PDE. I am considering the following PDE: $$ u_t+a^2u_{xx}=f\\ u(x,0)=u_0\\ u(x,t)|_\Gamma=u_g $$ That equation seems very like heat equation which only ...
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27 views

What is “Triangulable Triad”?

I am reading George W. Whitehead's Homotopy Theory; Corollary 1.0.2 mentioned the term "Triangulable triad" without definition. May I know how it is defined?
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Dyadic expansion: Definition of $F^{i-1}\omega$?

In Billingsley probability and measure under Dyadic expansion it is stated: dinfe a mapping $F$ from $\Omega=(0,1]$ into itself given by $$F\omega=\begin{Bmatrix} 2\omega & \mbox{if } 0< ...
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36 views

Motivation for defining a functions codomain

Why not always refer to a functions image, what is the point in specifying some super set of that image and then naming that super set the functions "codomain"? What does explicitly defining a set ...
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Definition review: how to make this geometric definition clearer?

In a paper I am writing, I rely on the following definition Given a geometric shape $C$ and a family of geometric shapes $S$, The division number of $C$ relative to $S$, denoted $DivNum(C,S)$, is ...
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101 views

Meaning of $A^A$

Let $A$ be a matrix. I know that there is definition for $A^k$ power of $A$, and $e^A$ expontential of $A$. Is there any meaning of $A^A$?
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Is there a term for an endomorphism defined up to conjugation by an automorphism?

Is there a standard term to designate the equivalence class of endomorphisms where two endomorphisms $\phi$ and $\psi$ are considered equivalent if there exists an automorphism $\alpha$ such that ...
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what are the basics of the Divergence operator?

Can you please help me with the basics about the divergence operator $div$. I don't know how to use it ? where to use it ?
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Definition of smooth (variety)

I don't understand the motivation for the definition of smoothness of a variety: A variety $V(f_1,...,f_m)$ in $n$-space is smooth iff $\mbox{rank}$ = $n-\mbox{dim} V$. Could you please give me ...
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79 views

Is it false that the complement of an open set is closed?

Let $f:\mathbb{C}\rightarrow \mathbb{C}$ be a continuous function. Let $Z(f)$ be the zero of $f$. Prove that $Z(f)$ is closed. This is one of problems in my mid-term exam. I have used ...
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Difference between a vector space and an algebra

I'm new to the subject of algebras and I would like to get a better understanding of what they are exactly. Am I right to say the follwing: ...
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29 views

Uniformly bounded function?

For all I know, the term "uniformly bounded" only makes sense when applied to a family of functions, sequences of functions, sets of functions with one parameter and such. But I've seen in several ...
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44 views

If a series converges does its sequence of partial sums converge?

By Definition, a series $\sum_{n=1}^\infty a_n$ converges if it's sequence of partial sums $S_n = \sum_{k=1}^n a_k$ converges. My question is, is the converse true? If $\sum_{n=1}^\infty a_n$ ...
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What is the definition of a norm in the context of rings?

On several places in ring-theory I encountered so-called 'norms'. For instance on integral domain $\mathbb{Z}\left[i\right]$ with prescription $a+bi\mapsto a^{2}+b^{2}$ where it also serves as a ...
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36 views

The term $rank$ in methematics

Reading wikipedia's disambiguation page about the "rank" word I see many concept of rank of many different matematical object. I only know about the rank of a graded poset and the rank of a set that ...
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60 views

what is the argument of 0?

When $z\neq 0$, $\arg z$ is defined to be the set $\{\theta \in \mathbb{R} : z=|z|e^{i\theta}\}$. What if $z=0$? Usually does one leave the argument of $0$ undefined? Or is $\arg 0 = \mathbb{R}$?
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About definition of “extended absolute value” (in $\overline{\Bbb{R}}=\Bbb{R}\cup\{+\infty,-\infty\}$)

Is correct following definition? Def.: Let be $a \in \overline{\Bbb{R}}=\Bbb{R}\cup\{+\infty,-\infty\}$, $$|a|=\begin{cases} |a|& \text{ if } a\in \Bbb{R} \\ +\infty & \text{ if } a \in ...
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31 views

Depth in acyclic graphs

I am struggling to understand a definition in a paper: Given a acyclic (directed!) graph $D=(V,E)$ we define a sequence $Q_i \subset V(D)$ of sets: $$Q_0 = \emptyset,$$ $$ Q_i \textrm{ is ...
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In the definition of NP, is it required to have polynomially bounded length of certificate?

So given the definition in our lectures, we were told that NP is defined as the set of languages $L$ s.t. there exist a polynomial time bounded Turing-acceptor M s.t. $L ={w: M accepts(w#c) for some ...
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42 views

Particular “upper bound” and “lower bound” of sequence

Is it possible to define the following? Let $a:\Bbb{N} \to \Bbb{R}$, and $M \in \Bbb{R}$, $M$ is ? if $$\exists n \in \Bbb{N}(\forall p \in \Bbb{N}(p\geq n \to a(p)\leq M))$$ Let $a:\Bbb{N} \to ...
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Continuous time Markov chains, how would this definition be expanded from time-homogeneous to time-inhomogeneous.

Below I have a picture of how we can view a continuous time Markov chain that is time-homogeneous. Now, I am wondering what happens when we have a inhomogeneous continuous Markov chain. I have ...
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24 views

Proper equality sign for boundary value definition

Say I have a function that needs to define a boundary condition, like $f(0) = A$. In this case it is usually fairly self evident from context, that this is a requirement that $f(0)$ needs to satisfy. ...
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Understanding definition of Full Covers

Let $[a, b]$ be a given closed, bounded interval and let $X$ be a subset of $[a,b]$. A collection $\textbf{C}$ of closed subintervals of $[a, b]$ is a full cover of $X$ if to each $x$ in $X$ there ...
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definition of a fiber bundle

I came across the definition of a fiber bundle in May's "A Concise Course in Algebraic Topology" (http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf, Chapter 7, section 4). There were two ...
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48 views

Definition of self-adjoint endomorphism

let $f \in End_K(E)$, and $g: (E \times E)\to \Bbb{R}^1$ a symmetric bilinear form positive definite, $f$ is self-adjoint endomorphism if $$\forall v,w \in E(g(f(v),w)=g(v,f(w)))$$ It is correct? ...
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Definition of $\partial^2{u}/ \partial {n^2}$

I want to solve a boundary value problem on a square, that one of the boundary condition is $$\frac{\partial^2{u}}{ \partial {n^2}}=0.$$ The test example that I want to solve is: $$\Delta^2u=f,$$ ...
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37 views

Question about the definition of a supremum

I want to know if the next definition is correct or I should fix it: Lets say that we have group $K$ and $a\in K$. Then, if $a\sup K$, for every $x\in K\Rightarrow x<a$? Or I shold cahge it to ...
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Modules,rings and definitions

Is there a source available with (almost) all definitions from ring and module theory,all in ONE place without theorems.There are books on module theory where are freely used unusual notions like ...
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Is there a class of schemes s.t. morphisms from a quasi-compact (-separated) scheme $X$ to any one of them is quasi-compact (-separated)?

The question is intend for both quasi-compact or quasi-separated notions, but let me elaborate on the details of quasi-compact only. Def: A scheme $X$ is said to be quasi-compact if any of its covers ...
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optimum lengths for a gauge block set

Has there been any mathematical study of the "optimum" lengths for a gauge block set? What do you call such a set of lengths? I'm looking for something analogous the way different ways of ...
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64 views

Maximum measurable collection and Caratheodory extension

Remember Caratheodory extension theorem: we get an outer measure by extending a premeasure over an algebra. We also have Caratheodory theorem, which give us a measure by restricting that outer measure ...
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Adjacency corner case

If two rectangles touch, but only touch at one of their corners (example, Rectangle A's upper-right corner is touch Rectangle B's lower-left corner), are they adjacent to one another? Why or why not?! ...
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Proper vs Subline Adjacency

Can someone explain to me the technical difference between "proper" and "subline" adjacency? The only definition I have is as follows: Adjacency may be proper or subline, where a subline adjacency ...
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What does it mean that solution of DE has ''n zeros''?

It appears in the definition of disconjugate DE: A linear DE with constant coefficents on a $t$-interval $I$ is said to be disconjugate on $I$ if no solution ($\neq 0$) has $n$ zeros on $I$. Thank ...
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definition of derived algebra $[L,L]$ of a Lie algebra $L$

Definition of derived algebra of a Lie algebra $L$ is given by linear span of commutators $[x,y]$ for $x,y \in L$. but here why do we take linear span and why cant we just consider collection of all ...
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102 views

Simplicial Homology Group, Delta set/complex

I'am confused. I'm trying to learn something about homology groups, but unfortunately I'm lost in some of the definitions/notations. Is there any difference between a delta set and a delta complex? ...
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33 views

Definition of restriction of relation

let be $\mathcal{R}$ a binary relation on $X$, and $\mathcal{T}$ a binary relation on $Y$, $\mathcal{T}$ is restriction of $\mathcal{R}$ if: 1) $Y \subseteq X$ 2) $\forall a,b \in Y( a T b ...
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66 views

Formal and general definition of natural domain (natural set) of a function

Can anyone give me a (as much as possible) formal and general definition of natural domain of a function? Let's say that a function is a triplet $(X,Y,f)$ where $f \subseteq X \times Y$ such that ...